Feeds:
Posts
Comments

Posts Tagged ‘Labyrinth’

An interesting labyrinth is reproduced in the book of Kern (fig. 200, p. 119)°. A drawing by Arabian geographer Al Qazvini in his cosmography completed in 1276 is meant to show the ground plan of the residence of the ruler of Byzantium, before the large city of Constantinople was built up.

This non-alternating labyrinth has 10 circuits and a unique course of the pathway. I will show this using the Ariadne’s Thread and the pattern. In my post “From the Ariadne’s Thread to the Pattern – Method 2” (see related posts, below), I have already described how the pattern can be obtained. When deriving the pattern I always start with a labyrinth that rotates clockwise and lies with the entrance from below. The labyrinth by Qazvini rotates in clockwise direction, however it lies with the entrance from above. Therefore I rotate the following images of the labyrinth by a semicircle so that the entrance comes to lie from below. So it is possible to follow the course of the pathway with the Ariadne’s Thread and in parallel see how this is represented in the pattern.

Four steps can be distinguished in the course of the pathway.

Phase 1

The path first leads to the 3rd circuit. The entrance is marked with an arrow pointing inwards. In the pattern, axial sections of the path are represented by vertical, circuits by horizontal lines. The way from the outside in is represented from above to below.

Phase 2

In a second step, the path now winds itself inwards in the shape of a serpentine until it reaches the 10th and innermost circuit. Up to this point the course is alternating.

Phase 3

Next follows the section where the pathway leads from the innermost to the outermost circuit whilst it traverses the axis. In order to derive the pattern, the labyrinth is split along the axis and then uncurled on both sides. As the pathway traverses the axis, the piece of it along the axis has to be split in two halves (see related posts below: “The Pattern in Non-alternating Labyrinths”). This is indicated with the dashed lines. These show one and the same piece of the pathway. In the pattern, as all other axial pieces, this is represented vertically, however with lines showing up on both sides of the rectangular form and a course similarly on both sides from bottom to top.

Phase 4

Finally the pathway continues on the outermost circuit in the same direction it had previously taken on the innermost circuit (anti clockwise), then turns to the second circuit, from where it reaches the center (highlighted with a bullet point).

Related Posts:

°Kern, Hermann. Through the Labyrinth – Designs and Meanings over 5000 Years. Munich: Prestel, 2000.

Read Full Post »

Once more: Type in Style

I have now needed three posts to attribute all labyrinth examples of this series to their types. Here I present the last part.

 


Examples in the Reims style

Reims 1

Reims 1

 

 

 

Type Reims


Reims 2

Reims 2

 

 

 

Type Reims


Reims 3

Reims 3

 

 

 

Type Chartres


Reims 4

Reims 4

 

 

 

Type Sneinton (labyrinth drawn faultily)


Reims 5

Reims 5

 

 

 

Type Saffron Walden (labyrinth drawn faultily)


Exemples in the Knidos Style

Knidos 1

Knidos 1

 

 

 

Type Knossos


Knidos 2

Knidos 2

 

 

Core-labyrinth of the type Rockcliffe Marsh, doublespiral-like mander (Erwin’s type 6 meander)


Knidos 3

Knidos 3

 

 

 

 

Cretan type


Knidos 4

Knidos 4

 

 

 

 

Type Otfrid


Other Examples

Andere 1

Other 1

 

 

 

Type Rockcliffe Marsh


Andere 2

Other 2

 

 

 

 

Cretan Type


Andere 3

Other 3

 

 

 

 

Cretan Type


Andere 4

Other 4

 

 

 

 

Type Al Qazwini


Andere 5

Other 5

 

 

 

Type Cakra Vyuh


Andere 6

Other 6

 

 

 

Type Liger


Andere 7

Other 7

 

 

 

Type Ely


Andere 8

Other 8

 

 

 

Type Kieser


Andere 9

Other 9

 

 

 

 

Type Gent


We can see here a similar result as in the two previous posts. The 18 examples belong to 14 different types.
What can be seen here also is, that in some labyrinths the pattern may be difficult to obtain (type Liger, type Ely, type Kieser, type Gent). I do not explain this further here because this is beyond the space of this post.

The types used

Related posts:

Read Full Post »

Type in Style (continued)

Here I will now attribute the labyrinth examples from my post Type or Style / 9 to types of labyrinths. These examples were grouped in styles.

The examples in the classical style

Klassisch 1

Classical 1

 

 

 

Type Löwenstein 3


Klassisch 2

Classical 2

 

 

 

Type Löwenstein 5a


Klassisch 3

Classical 3

 

 

 

Cretan Type


Klassisch 4

Classical 4

 

 

 

 

Type Löwenstein 9b


Klassisch 5

Classical 5

 

 

 

Type Hesselager


Klassisch 6

Classical 6

 

 

 

Type Tibble


The examples in the concentric style

Konzentrisch 1

Concentric 1

 

 

 

 

Cretan Type


Konzentrisch 2

Concentric 2

 

 

 

Type Hesselager


Konzentrisch 3

Concentric 3

 

 

 

 

Type Otfrid


Konzentrisch 4

Concentric 4

 

 

 

 

Type Chartres


Konzentrisch 5

Concentric 5

 

 

 

 

Type Gossembrot 51r


Konzentrisch 6

Concentric 6

 

 

 

Type Münster


The examples in the Man-in-the-Maze style

MiM 1

MiM 1

 

 

 

Cretan Type


MiM 2

MiM 2

 

 

 

 

Type Pima


MiM 3, MiM 4: no true labyrinths


The examples in the Chartres style

Chartres 1

Chartres 1

 

 

 

Type Chartres


Chartres 2

Chartres 2

 

 

 

Type Trinity


Chartres 3

Chartres 3

 

 

 

 

Type St. John


Chartres 4

Chartres 4

 

 

 

 

Type Petit Chartres


Chartres 5: no true labyrinth

 

Chartres 6

Chartres 6

 

 

 

Type Grey’s Court


In order not to overload this post I interrupt here and present the remaining examples in my next post.

Among the examples here, there are also figures, that are no unicursal labyrinths in the strict sense. For reasons of space I do not explain this further here but will come back to it later.

The types used:

Read Full Post »

Summary

As is the case with the labyrinth itself and the seed pattern, there are also two representations of the rectangular form: this can be represented either with the walls or with the Ariadne’s Thread. In addition, there are two methods to obtain the rectangular form and therefore two versions of it. Ill. 1 summarizes this with the example of my demonstration labyrinth.

L:SP:P Rep

Illustration 1. Overview

This illustration shows on the first line the labyrinth (figures 1), on the second line the seed pattern (figures 2), on the third line the rectangular form obtained with method 1 (figures 3) and on the bottom line the rectangular form obtained with method 2 (figures 4). Each of these are shown in the representation with the walls (left figures a) and with the Ariadne’s Thread (right figures b).

  • When we speak of a „labyrinth“ we usually mean the labyrinth in its representation with the walls. This is shown in fig. 1 a. But also the representation with the Ariadne’s Thread is in widespread use and generally well known (fig. 1 b). This is usually simply referred to as the Ariadne’s Thread.
  • Fig. 2 a shows the seed pattern for the walls, fig. 2 b the seed pattern for the Ariadne’s Thread. As Erwin and I have written so much about this in recent posts, I don’t elaborate more on it here.
  • If we start from the labyrinth (fig. 1 a) or from the Ariadne’s Thread (fig. 1b) and apply method 1, we will as a result obtain the rectangular forms shown in line 3. Thus, there exists a rectangular form for the walls (fig. 3a) as well as for the Ariadne’s Thread (fig. 3b).
  • If we apply method 2 this results in the rectangular forms of line 4. These are the same as the figures on line 3, although rotated by half the arc of a circle.

For what I termed “rectangular form” here, in the literature we can find also the terms „compression diagram“ or „line diagram“ or else. And, most often, we will encounter rectangular forms for the walls obtained with method 1, i.e. figures like fig. 3 a.

RF BM M1

Illustration 2. Figure 3a

I, however, always use the rectangular form for the Ariadne’s Thread. This is the simpler graphical representation. Furthermore, I use the version obtained with method 2, as the result can be read from top left to bottom right, what corresponds better with the way we are used to read. This figure (e.g. fig. 4 b), the rectangular form for the Ariadne’s Thread obtained with method 2, is what I refer to as the pattern.

RF AF M2

Illustration 3. Figure 4b

Related posts:

Read Full Post »

Method 2

In the last post I have shown a first method of how to transform the Ariadne’s Thread into the rectangular form. For this, one of the halves of the axis was fixed and the other rotated by a full turn along the circuits. This resulted in the pattern with the entrance on bottom right and the center on top left. Here I will show a second method.

Lage KS

Figure 1. Ariadne’s Thread and Situation of the Seed Pattern

We start from the same baseline situation as in method 1. The labyrinth is presented with it’s Ariadne’s Thread with the entrance at the bottom and in clockwise rotation (fig. 1).

Muster Meth2

Figure 2. Rotating Both Halves of the Axis Upwards by Half a Circle

In method 2, however, each half of the axis is rotated by half a turn along the circuits (fig. 2).

Both halves then meet on top of the circuits. Perhaps, this figure shows even better, how by flipping up both ends of the axis the circuits are shortened from full circles to short lines.

Muster Erg2

Figure 3. Result: Pattern with Entrance on Top Left and Center on Bottom Right

After straightening-out the result shows the same pattern as in method 1. However it now lies with the entrance on top left and the access to the center on bottom right.

In both methods we started from the same labyrinth in the same basic situation. Both methods lead to the same pattern. However, in method 1, the pattern lies with the entrance on bottom right and the center on top left. In method 2 this is rotated by 180 degrees so that the entrance lies on top left and the center on bottom right. This orientation of the pattern corresponds better with the way we are used to read. For that reason, I prefer method 2.

Related posts:

Read Full Post »

Method 1

In my last post I have shown how the seed pattern can be transformed into the pattern. The same result can be obtained by transforming the Ariadne’s Thread into the rectangular form.

Lage KS

Figure 1. Ariadne’s Thread and Situation of the Seed Pattern

Fig. 1 shows the Ariadne’s Thread of my demonstration labyrinth with the seed pattern highlighted. In addition the situation of the entrance (arrow) and of the center (bullet point) is indicated.

AF-M

Figure 2. Rotating the Right Half of the Axis…

In fig. 2 we now fix the left half of the axis and rotate the right half anticlockwise a full turn along the circuits. By this, the circuits are continually shortened. Immediately before the right half reaches the left half of the seed pattern on its opposite side, the circuits have reduced to very short lines. But, as can be seen, it is really the circuits of the labyrinth, that connect the ends of both halves of the seed pattern.

Mäander_Meth1

Figure 3. … Until it Meets the Left from the Other Side

At the point where both halves meet each other, these remaining pieces of the circuits disappear. In lieu of them the straight of the meander appears. This is composed of the outer vertical lines of the original auxiliary figure of the seed pattern.

Therefore, it is absolutely justified to straighten-out the meander at the point where it intersects with the vertical straight. The lines that connect the ends of the seed pattern really represent the circuits of the labyrinth.

In fig. 3 we have now generated the meander starting from the Ariadne’s Thread, fixing one half of the seed pattern and rotating the other by a full turn. I refer to this way of generating the pattern as method 1. I had fixed the left half and rotated the right half.

Muster Meth1b

Figure 4. Rotating the Left Half of the Axis by a Full Turn

Fig. 4 shows that we could also fix the right half and rotate the left. In the result, this makes no difference.

Muster Erg1

Figure 5. Result: Pattern with Entrance on Below Right and Center on Top Left

The result of this method is in both cases the same meander that is straightened-out to the pattern as described previously.

Important: Please notice, that after this transformation, in the pattern the entrance lies on the bottom right and the center on top left. This result is against our spontaneous intuition and also contradicts with how we are used to read. It is a result of the applied method 1.

Related posts:

Read Full Post »

In my last post I have shown that a seed pattern can also be drawn in labyrinths with multiple arms (see related posts). With the seed pattern for the Ariadne’s Thread it is possible to connect the arms in the shape of a flower (or a propeller). For each arm, a separate petal is needed.

Figure 1. Borderlines

Figure 1. Borderlines

These auxiliary figures can all be drawn in one continous  line.

Figure 2. How to Draw the Borderline

Figure 2. How to Draw the Borderline

Fig. 2 illustrates this with a three-arm labyrinth. Other flowers with multiple arms can be drawn the same way.

Each of the petals contains a part-seed pattern. I will show this using an example of one of my five-arm labyrinth designs. I choose the design KS 2-3 that has been installed as a temporary labyrinth on the square of Magdeburg cathedral. This labyrinth can be actually seen in the header of this blog. Otherwise there is an image of it here.

Figure 3. KS 2-3

Figure 3. KS 2-3

Fig. 3 shows the labyrinth in a drawing by Erwin of the walls and with the Ariadne’s Thread (red) inscribed.

Figure 4. Seed Pattern for the Ariadne's Thread

Figure 4. Seed Pattern for the Ariadne’s Thread

In fig. 4 the seed pattern for the Ariadne’s Thread is shown per se and completed to the whole Ariadne’s Thread. In order to complete this seed pattern, for each circuit 10 ends have to be connected with five segments of a circuit. Evidently, with an increasing number of arms, the seed pattern comes to look closer to the entire labyrinth. The part-circuits are shortened in relation to the part-seed patterns.

The seed pattern has first and most often been published for the walls of the Cretan type labyrinth. Also for several other one-arm labyrinths, seed patterns have been published. Therefore, the seed pattern does not constitute a characteristic attribute of the Cretan type labyrinth. It is not even a characteristic specific for one-arm labyrinths alone.

However, the use of seed patterns in labyrinths with multiple arms is of minor practical importance. The original purpose and meaning of the seed pattern was, that it enables us to capture the essential of a labyrinth with a simple memorable system of lines that allows us to generate the labyrinth straight away. And this applies best to the seed patterns of the Cretan type labyrinth and its relatives of the vertical line.

Figure 5. Seed Patterns of the Cretan Labyrinth and it's Relatives of the Vertical Line

Figure 5. Seed Patterns of the Cretan Labyrinth and it’s Relatives of the Vertical Line

Fig. 5 shows the seed patterns for the walls in the first column and for the Ariadne’s Thread in the second column. The corresponding types of labyrinths are on

  • row 1: Löwenstein 3
  • row 2: The Cretan
  • row 3: Hesselager
  • row 4: Tibble

 

Related Posts

 

Read Full Post »

Older Posts »

%d bloggers like this: